acoustic metamaterials with circular sector cavities and...

Acoustic metamaterials with circular sector cavitiesand programmable densities

W. Akl and A. ElsabbaghFaculty of Engineering, Ain Shams University, Cairo, 11517, Egypt

A. BazMechanical Engineering Department, University of Maryland, College Park, Maryland 20742

(Received 6 July 2011; revised 23 April 2012; accepted 26 April 2012)

Considerable interest has been devoted to the development of various classes of acoustic metamaterials

that can control the propagation of acoustical wave energy throughout fluid domains. However, all the

currently exerted efforts are focused on studying passive metamaterials with fixed material properties.

In this paper, the emphasis is placed on the development of a class of composite one-dimensional

acoustic metamaterials with effective densities that are programmed to adapt to any prescribed pattern

along the metamaterial. The proposed acoustic metamaterial is composed of a periodic arrangement of

cell structures, in which each cell consists of a circular sector cavity bounded by actively controlled

flexible panels to provide the capability for manipulating the overall effective dynamic density. The

theoretical analysis of this class of multilayered composite active acoustic metamaterials (CAAMM) is

presented and the theoretical predictions are determined for a cascading array of fluid cavities coupled

to flexible piezoelectric active boundaries forming the metamaterial domain with programmable

dynamic density. The stiffness of the piezoelectric boundaries is electrically manipulated to control the

overall density of the individual cells utilizing the strong coupling with the fluid domain and using

direct acoustic pressure feedback. The interaction between the neighboring cells of the composite

metamaterial is modeled using a lumped-parameter approach. Numerical examples are presented to

demonstrate the performance characteristics of the proposed CAAMM and its potential for generating

prescribed spatial and spectral patterns of density variation. VC 2012 Acoustical Society of America.

[http://dx.doi.org/10.1121/1.4744936]

PACS number(s): 43.40.At, 43.40.Fz, 43.50.Fe [ANN] Pages: 2857–2865

I. INTRODUCTION

Metamaterials, either acoustic or electromagnetic, have

recently attracted the focus of many researchers as a new

technique for achieving wave propagation patterns, which

are impossible to realize using regular composite materi-

als.1–3 Developing acoustic or electromagnetic metamateri-

als is physically analogous to engineering periodic material

structure using the inclusions of small inhomogeneities to

enact effective macroscopic behavior.4 Acoustic metamateri-

als are therefore considered as those material structures,

rather than compositions, that are designed and artificially

fabricated to control, guide, and manipulate sound in the

form of sonic, infrasonic, or ultrasonic waves, as these might

occur in gases, liquids, and solids. The hereditary line into

acoustic metamaterials follows from the theory and research

in electromagnetic metamaterials as first developed by Pen-

dry5 who has proven that using negative refractive index

would result in a perfect lens. In 2006, Pendry et al.6 were

the first to present the transformation-based solutions to the

Maxwell’s equations, which have proven to yield a general

method for rendering arbitrarily sized and shaped objects

electromagnetically invisible. This was based on the invari-

ance of the Maxwell’s equations under coordinate transfor-

mation.7,8 Further, with acoustic metamaterials, sonic waves

can now be extended to the negative refraction domain9.

Control of various forms of sonic waves mostly requires

controlling of the bulk modulus B and the density q, which

are counterparts to the electromagnetic permittivity and per-

meability. The speed of sound, on the other hand, being

totally dependent on B and q, is analogous to the refractive

index in electromagnetic domain. It is therefore due to the

analogy to the wave equation in electromagnetic, researchers

have developed the theoretical foundation of acoustic meta-

materials,10,11 where their focus was directed toward various

wave propagation control applications with special interest

in acoustic cloaking rendering objects acoustically invisible.

Several attempts to theoretically and physically control

the bulk modulus and density of different material composi-

tions and structures have been reported, aiming eventually at

controlling the acoustic wave propagation. On the path to

control the bulk modulus, two approaches have been

reported in literature; the first was by combining two differ-

ent isotropic materials in a composite pattern to yield aniso-

tropic properties that can influence the spatial wave

propagation patterns.12–14 In the second mechanism, acoustic

impedance mismatch is introduced along the path of wave

propagation by integrating flexible sections into the rigid-

walled waveguides in order to vary the speed of sound and

effective bulk modulus at these sections.15–17 Manipulation

of the material density, on the other hand, has also been

reported using two different approaches; the first was by

combining two different materials with different densities in

a specific spatial arrangement that would yield a homoge-

nized value of the density all over the domain. Such an

approach was adopted by18–20 using phononic crystals. In

J. Acoust. Soc. Am. 132 (4), Pt. 2, October 2012 VC 2012 Acoustical Society of America 28570001-4966/2012/132(4)/2857/9/$30.00

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 50.81.134.248 On: Fri, 11 Sep 2015 23:01:24

the second approach, the concept of dynamic density,

which depends on the neighboring material stiffness, was

implemented. This approach was manifested in the attempts

of synthesizing prescribed dynamic acoustic densities in

fluid domains by introducing lattice systems of mass-in-

mass units.21–24 These attempts merely focused on intro-

ducing negative effective density motivated by the mathe-

matical analogy between acoustic and electromagnetic

waves, where the theoretical possibility of having negative

electromagnetic permittivity and permeability was intro-

duced by Pendry.5

In all these studies the focus has been placed on passive

metamaterials with fixed material properties, which has con-

siderably limited the applicability of this new generation of

structured metamaterials in wave propagation control due to

the narrow operating bandwidth in the vicinity of the internal

cell structure resonances. Very few researchers have tackled

the idea of having programmable material properties (bulk

modulus and/or density). Baz25,26 studied the concept of the-

oretically realizing programmable density using a periodic

structure that contains actively controlled piezoelectric ele-

ments coupled with fluid domain to control the effective

dynamic density in the composite structure. Akl and Baz27

have also applied the same concept in controlling the effec-

tive bulk modulus of the composite domain. In their devel-

opment, the authors have, however, limited their model to

straight one-dimensional structures. In this paper, the empha-

sis is placed on the development of a new class of one-

dimensional composite circular sector-type acoustic meta-

materials with tunable effective densities, which can be tai-

lored to have increasing or decreasing variation along the

material volume. Due to the proposed geometry, which is

more oriented toward applications involving cylindrical

domains such as acoustic cloaking for example, the wave

fronts introduced follow the same physics of two-

dimensional circular waves, such as amplitude dependence

on the radial distance from the sound source. Such depend-

ence introduces more complexity in calculating the acoustic-

electrical circuit analogy manifested in the expressions of

the capacitors and inductors representing the fluid domain

compliance and mass, respectively, which to the authors’

knowledge has not been treated yet. A piezoelectric sector

element coupled with a circular sector acoustic cavity is

introduced to form a homogenized periodically structured

sector-shaped acoustic metamaterial. The effective dynamic

density is controlled using the piezoelectric ingredient in the

composite structure. A detailed theoretical analysis is intro-

duced to present a lumped-parameter model of the developed

metamaterial. Acoustic-electrical circuit analogy is used to

model the cavity characteristics, as this approach has proven

to be effective, provided the overall dimensions of the cavity

are much smaller than the wavelengths of the acoustic waves

passing through. A set of cascading “cells” of the developed

metamaterial cell is also introduced showing the capability

of such configuration to control the densities along the path

of the wave propagation. In addition, the necessary precau-

tions to eliminate the instabilities are introduced. These

instabilities occur in the piezoelectric element due to the

active control that might exceed the buckling limit leading

to entire damage in the integrity of the composite metamate-

rial structure.

II. ELECTRIC ANALOGY OF CIRCULAR SECTORACOUSTIC CAVITIES

In order to derive the acoustic-electric circuit analogy to

circular sector acoustic cavities, a brief summary of the

acoustic-electric straight cavity is introduced. Euler and con-

tinuity equations are presented, which are later implemented

in modified form to capture the divergence effect of the cav-

ity cross section on the wave equation in sector acoustic

cavities.

A. Straight cavity

A straight acoustic cavity with uniform cross-sectional

area Ast and length lf, subject to acoustic pressure drop Dpresulting in volumetric flow rate dQ, is considered. The fluid

considered inside the cavity is characterized with static fluid

density qf and bulk modulus Bf. The value of dQ depends on

three major forces; inertia forces due to the mass of the fluid

inside the cavity, elastic forces due to the “stiffness” of the

entrapped fluid volume, and finally friction and damping

forces, which are ignored in the current analysis.

1. Inertia forces

Newton’s second law necessitates that the net force due

to the pressure difference Dp equals the rate of change of

fluid momentum as given by

AstDp ¼ qf lf Ast

dv

dt; (1)

where v is the particle velocity. For a uniform velocity along

the cavity length, dQ=dt ¼ Astðdv=dtÞ, the inertial imped-

ance of the straight cavity after Laplace transformation is

given by

DPðsÞQðsÞ ¼

qf lf

Ast

s ¼ LFs ¼ ZLFðsÞ: (2)

2. Elastic forces

The second type of forces is due to the stiffness of the

fluid domain. A pressure drop of Dp across the cavity will

cause the fluid to be “strained” due to change in its volume

Vol by an amount of d(Vol) such that eV¼�d(Vol)/Vol, and

as the bulk modulus Bf is defined as the difference in pres-

sure, which yields a unit volume strain, Dp¼�Bf d(Vol)/

Vol. The change in volume d(Vol) along a period of time dtis defined as: d(Vol)¼�

ÐQ dt. Hence, and applying Laplace

transformation to Dp, the elastic and total impedance for the

straight acoustic cavity ZCF and Zt are calculated as

ZCFðsÞ ¼DPðsÞQðsÞ ¼

1

CFs¼ 1

s

Bf

Astlf(3)

ZtðsÞ ¼ ZLFðsÞ þ ZCFðsÞ ¼qf lf s

Aþ Bf

Astlf s: (4)

2858 J. Acoust. Soc. Am., Vol. 132, No. 4, Pt. 2, October 2012 Akl et al.: Active acoustic metamaterials


B. Circular sector acoustic cavity

A sector of the fluid domain with point source and

acoustic pressure wave propagating in the radial direction is

as shown in Fig. 1(a). The sector cavity shown, expands

from r¼ r1 to r¼ r2, which includes a central angle c. A slice

of thickness dr is considered for force balance and mass flow

rate calculations.

1. Inertia forces

Considering the infinitesimal slice of thickness dr in the

sector cavity, Newton’s second law necessitates that AðrÞdp¼ qf AðrÞðdv=dtÞdr, where A(r)¼ cHr, H is the cavity depth.

As dQ=dt ¼ AðrÞðdv=dtÞ and integrating (r1! r2), the iner-

tial impedance term is defined as

ZLF ¼DPðsÞQðsÞ ¼ s

qf

Hcln

r2

r1

; (5)

where in the limit case (c ! 0), ZLF reduces to the form for

straight cavity (limc!0ZLFðsÞ ¼ sðqf ðr2 � r1Þ=Hb1Þ), where

b1 is the chord lengths of the inner circular sector.

2. Elastic forces

For sector cavity, a different approach is implemented

in order to calculate the effect of the elastic forces on the

acoustic cavity as follows:ðr2

r1

Fdr ¼ DPE; (6)

where F is the elastic force inside the sector cavity and DPE

is the change in potential energy. Rewriting Eq. (6) such thatÐ r2

r1AðrÞTdr ¼ DpDVol, where A(r) is the cross-sectional

area of the sector cavity at radius r, T is the normal stress,

Dp is the pressure difference across the sector cavity, and

substituting for T with S=SE, where S is the mechanical strain

and sE is the mechanical compliance, dividing by DVol

inside the integral and completing the integration while

replacing 1/sE with fluid bulk modulus Bf results in

Dp ¼ðr2

r1

AðrÞlf Bf

Hcrl2f

dr ¼ DVol Bf

Hcl2fln

r2

r1

(7)

from which the elastic and total impedances ZCF and Zt, as

shown in Fig. 1(b) are calculated as

ZCFðsÞ ¼DPðsÞQðsÞ ¼

1

CFs¼ 1

s

Bf

Hcl2fln

r2

r1

; (8)

ZtðsÞ ¼ ZLFðsÞ þ ZCFðsÞ ¼ sqf

Hcln

r2

r1

þ 1

s

Bf

Hcl2f

lnr2

r1

:

(9)

C. Dynamic density for sector acoustic cavity

Combining both the “inertia” and stiffness impedances

and substituting Bf¼qfc2f, where cf is the speed of sound in

the fluid domain, the dynamic equation for the circular sector

acoustic cavity is given as

Dp ¼ �qf sQ

Hcln

r2

r1

þc2

f

l2f

lnr2

r1

1

s2

!: (10)

For very small cavities, ðr2=r1 ¼ 1þ a; a� 1Þ and defin-

ing Q¼AAve� � and qeff:�Dp=lf ¼ �qeffðdv=dtÞ

�, the rel-

ative density of the acoustic cavity is defined as

qeff

qf

¼ AAve

Hclfln

r2

r1

þAAvec2

f

Hcl3f s2ln

r2

r1

!; (11)

which, for a straight cavity, would converge to the form

given by Baz.25

D. Dynamic density for sector acoustic cavity coupledwith flexible diaphragm

Considering a sector acoustic cavity coupled to a flexi-

ble diaphragm as shown in Fig. 2(a) with its analogous

lumped parameter electric circuit shown in Fig. 2(b), the

flexible diaphragm, coupled with the acoustic sector cavity,

represents a composite material that consists of two compo-

nents characterized with different densities (qf, qd), bulk

moduli (Bf, Bd), and thicknesses (lf¼ r2� r1, ld¼ r3� r2) for

the fluid and flexible diaphragms, respectively. Applying

electrical circuit analogy to the acoustic cavity, the overall

circuit impedance (Zt) is calculated by summing up the dif-

ferent impedances representing the different inductors and

capacitors in the circuit as follows:

ZLF ¼ LFs ¼ qf

1

Hcln

r2

r1

s; (12)

ZLD ¼ LDs ¼ qd

1

Hcln

r3

r2

s; (13)

ZCF ¼1

CFs¼ 1

s

Bf

Hcl2f

lnr2

r1

; (14)

ZCD ¼1

CDs¼ 1

s

Bd

Hcl2dln

r3

r2

; (15)

Zt ¼ðZCD þ ZLDÞZCF

ZCD þ ZLD þ ZCFþ ZLF: (16)FIG. 1. (Color online) (a) Geometry of the acoustic sector cavity, (b) electric

circuit analogous to the acoustic sector cavity.

J. Acoust. Soc. Am., Vol. 132, No. 4, Pt. 2, October 2012 Akl et al.: Active acoustic metamaterials 2859


Substituting Eqs. (12)–(15) into Eq. (16) and rearranging,

results in the acoustic impedance Zt for the two-material

composite cavity system as follows:

Zt ¼Bf

sHc

s2 lnr2

r1

c2f

þBd 1þ s2l2

d

c2d

� �ln

r3

r2

Bf l2d þ Bdl2

f 1þ s2l2dc2

d

� �0BBB@

1CCCA: (17)

Referring to Euler’s Equation for acoustic domains,

Dp ¼ðr3

r1

dp ¼ðr3

r1

qeff

AðrÞdQ

dtdr ¼ qeff

HcdQ

dtln

r3

r1

;

and applying Laplace transformation, the effective dynamic

density of the composite domain qeff is eventually defined as

qeff ¼Bf

s2 ln r3

r1

s2 lnr2

r1

c2f

þBd 1þ s2l2

d

c2d

� �ln

r2

r3

Bf l2d þ Bdl2f 1þ s2l2

d

c2d

� �0BBB@

1CCCA:

(18)

Equation (18) reveals that the effective dynamic density for

the composite material is negative and approaches that of

the fluid domain at higher frequencies as shown in Fig. 3,

where the fluid domain modeled was water (qf¼ 1000 kg/m3,

Bf¼ 2.25� 109 Pa) coupled with a thin steel diaphragm

(qd¼ 7800 kg/m3, Bd¼ 210� 109 Pa), which is typical behavior

as reported by Baz.26 Due to the dependence of the effective

dynamic density, for such composite system, on the effective

bulk modulus of the two-material homogenized system, it is tar-

geted to develop a methodology of varying the bulk modulus of

the flexible diaphragm in order to yield a “programmable”

effective density of the homogenized system that can assume

any prescribed value over a wide frequency range.

E. Dynamic density for sector acoustic cavity coupledwith piezoelectric diaphragm

Consider the sector acoustic cavity coupled with a flexi-

ble diaphragm shown in Fig. 2(a), where the flexible

diaphragm is replaced with a piezoelectric one. The basic

constitutive equation for a piezoelectric material is given by

SD

� �¼ sE d

d e

� TE

� �; (19)

where S¼ strain, D¼ electric displacement, T¼ stress,

E¼ electrical field, sE¼ compliance, d¼ piezoelectric strain

coefficient, and e¼ permittivity of piezoelectric material.

Equation (19) can be rewritten as

DVol

q

� �¼ CD dA

dA 1=zps

� Dpp

Vp

� �; (20)

where DVol¼ change in diaphragm volume, q¼ electrical

charge, Dpp and Vp are the acoustic pressure and voltage dif-

ference across the piezoelectric diaphragm. A similar deriva-

tion was presented by Prasad et al.28 However, in the current

derivation the electrical free capacitance of the piezoelectric

material Zp presented by Prasad et al. is replaced with an

equivalent circuit that comprises the capacitance of the piezo-

electric diaphragm Cp combined with in-parallel inductance

FIG. 2. (Color online) (a) Geometry

and (b) analogous circuit of acoustic

sector cavity coupled with flexible

diaphragm.

FIG. 3. (Color online) Dynamic effective density for an acoustic sector cav-

ity coupled with flexible diaphragm.



Lp and in-series capacitance Cs to allow for stability measures

of the active element as will be demonstrated later,

Zp ¼Lps

1þ LpCpCss2

Cp þ Cs

: (21)

1. Calculation of Cp

To calculate the value for Cp, the electric field E across

the piezoelectric sector element must be calculated. Hence, a

Gaussian cylinder of radius r and thickness H is constructed

ðÞ

E dA ¼ EÞ

dAÞ. The ends of the cylinder are normal to the

field and do not contribute to the integral. The only part that

contributes to the electric field is the sector confined by the

angle c, which area is calculated asÞ

dA ¼ crH. As the inte-

gral form of Gauss law is defined asÞ

E dA ¼P

q=e, wherePq (or simply q) is the total charge on the curved sector ele-

ment (q¼ eEcrH), moving a positive test charge between the

sector boundary surfaces, ranging from inner radius r2 to outer

radius r3, requires applied work equal to the change in the

electric potential energy DEPE ¼Ð r3

r2Fdr ¼

Ð r3

r2qEdr ¼ qVp,

where Vp ¼Ð r3

r2ðq=ecrHÞdr ¼ ðq=ecrHÞ ln ðr3=r2Þ, is the

potential difference between the two piezoelectric sector

sides, yielding an expression for Cp ¼ ecH= ln ðr3=r2Þ.

2. Calculation of CD and dA

To calculate the value for CD and dA as in Eq. (20), the

effect of the electric-mechanical coupling in the piezoelectric

materials on the total potential energy stored in the piezoelec-

tric sector element is calculated using Eq. (6), where, F in this

case is the elastic force inside the sector cavity due to both the

mechanical strain and the electrical field and DPE is the change

in potential energy such thatÐ r2

r1AðrÞTdr ¼ DpDVol. Using the

expression for the total stress inside the piezoelectric sector

T¼ (S� dE)/sE extracted from Eq. (19) and replacing sE with

1/Bd (bulk modulus of piezoelectric sector diaphragm), the

acoustic pressure difference across the sector is calculated as

Bd

cHl2d

ðr3

r2

DVol

rdr � cHldd

ðr3

r2

Edr ¼ Dpp: (22)

Integrating Eq. (22) yields an expression for Dpp ¼ ðBd=cHl2dÞDVol lnðr3=r2Þ þ cHlddVp, which can be rearranged to calculate

the values of CD ¼ ðcHl2d=BdÞ ln r3=r2 and dA ¼ dcHld=ln ðr3=r2Þ. The electrical circuit analogous to the coupled fluid-

piezoelectric sector cavities is hence as illustrated in Fig. 4. The

parameter /¼�dA/CD/, also called the transformer turns ratio

in the equivalent circuit representation, is the electroacoustic

transduction coefficient.

Using the piezoelectric diaphragm as a self-sensing ac-

tuator, then the second row of Eq. (20) gives, for a short-

circuit piezosensor, the electric charges in the piezoelectric

diaphragm q¼ dADpp. In which case, the voltage Vp applied

to the diaphragm can be generated by a direct feedback of

the charge q such that Vp¼�GdADpp, where G¼ feedback

gain. Then, the first row of Eq. (20) yields

DVol ¼ ðCD � d2AGÞDpp ¼ CDCDpp; (23)

where CDC¼ closed-loop compliance of the piezoelectric di-

aphragm. Figure 5 displays the corresponding electrical cir-

cuit analogous of the acoustic cavity integrated with a

closed-loop piezoelectric diaphragm characterized with

overall impedance given by

Zt ¼ðZ0p þ ZCDC þ ZLDÞZCF

Z0p þ ZCDC þ ZLD þ ZCFþ ZLF: (24)

Recalling the identities defined in Eqs. (12)–(15), and noting

that Bp ¼ ðHcl2d=CDCÞ ln r3=r2, the total impedance Zt and

the effective density qeff of the coupled composite sector

cavity may be expressed as

Zt ¼Bf ln

r3

r2

Hl2dsZ0pcþ ðBp þ qdl2

ds2Þ ln r3

r2

� �þ s2qf ln

r2

r1

Hl2dl2f sZ0pcþ ðBf l2d þ Bdl2

f þ qdl2dl2f s2Þ ln r3

r2

� �

Hsc Hl2dl2f sZ0pcþ ðBf l2d þ Bdl2f þ qdl2dl2

f s2Þ ln r3

r2

� � ; (25)

qeff ¼Bf ln

r3

r2

�Hl2

dsZ0pcþ ðBp þ qdl2ds2Þ ln r3

r2

þ s2qf lnr2

r1

Hl2dl2

f sZ0pcþ ðBf l2d þ Bdl2

f þ qdl2dl2f s2Þ ln r3

r2

� �

s2 Hl2dl2f sZ0pcþ ðBf l2d þ Bdl2f þ qdl2

dl2f s2Þ ln r3

r2

� �ln

r3

r1

: (26)

FIG. 4. Analogous circuit of an acoustic fluid domain coupled with open-

loop piezoelectric diaphragm.



After rearrangement of Eq. (26), the bulk modulus of the pie-

zodiaphragm Bp and the control gain needed to realize speci-

fied value for qeff can be represented as a frequency

dependent function of Z0p as follows:

Bp ¼ s2l2d

Bf qeff lnr3

r1

� qf lnr2

r1

� �

l2f s2 qf lnr2

r1

� qeff lnr3

r1

� �þ Bf ln

r3

r2

0BB@

�HZ0pc

lnr3

r2

� qd

1CA; (27)

G ¼ Hcl2d

d2A ln

r3

r2

1

Bd� 1

Bp

� �: (28)

A numerical simulation of a sector-shaped composite cavity

using the developed model is developed and the frequency

dependent piezoelectric bulk modulus Bp, feedback control

gain G and resultant effective density qeff for the composite

system for target relative density values (qr¼ qeff/qf¼ 20)

and (qr¼ 0.05) are presented in Fig. 6, which were devel-

oped using the values listed in Table I and specific values of

Cs¼ 5� 10�10 Farad and LP¼ 0.02 H. It is important, how-

ever, to optimize the values of Cs and Lp to obtain the largest

bandwidth with feasible control gain values. Hence, a sensi-

tivity analysis is needed to determine the stability bandwidth

of the piezodiaphragm without reaching a bucking state of

the diaphragm material due to the applied control voltage.

III. CASCADING COMPOSITE CAVITIES WITHVARIOUS PROGRAMMABLE DENSITIES

In the previous section, a single cavity coupled with a

piezoelectric diaphragm was considered. The effective

dynamic density of the composite domain was proven to be

controllable by varying the dynamic bulk modulus of the

piezoelectric diaphragm to yield a frequency-dependent

dynamic density as required. In order to expand the applic-

ability of the introduced methodology, a set of cascading and

coupled composite metamaterial cells are modeled such that

each is programmed to yield different dynamic density

value. This setup is very important in the path to realize

phenomena such as acoustic cloaking, where fluid layers

circumscribing a solid cylinder, for example, can yield the

cloaking phenomenon, provided that their densities and bulk

modulus take a prescribed set of values as demonstrated by

Cheng et al.29. The schematic presentation of the cascading

cavity cells and the corresponding analogous electric circuits

are as shown in Fig. 7.

FIG. 5. Analogous circuit for an acoustic fluid domain coupled with closed-

loop piezoelectric diaphragm.

FIG. 6. (Color online) (a) Bulk modulus of piezodiaphragm, (b) control volt-

age, and (c) relative density for qr¼ 20 and 0.05.



In this system, the last cell represents the cell subject to

external pressure excitation, while the first one is the end of

the chain of the cascading cells. Due to the coupling nature

of the cells, a recursive solution pattern has to be adopted in

order to calculate the different values of the control gains

G(i) needed to alter the effective bulk modulus of the piezo-

electric diaphragms coupled to each cell Bp(i) to ensure a pre-

scribed set of effective densities within the cascading

coupled cells.

The recursive approach starts with the first cell, which is

coupled from one side only to the rest of the metamaterial do-

main. Using the electrical analogy, the overall impedance Z(1)

of this cell can be calculated as a function of the piezoelectric

diaphragm Bulk modulus Bp(1) and the required effective den-

sity qeff(1), as given by Eqs. (27) and (28). Once calculated,

this impedance is shunted to the circuit representing the rest

of the chain. Hence the second cell, loaded with the imped-

ance of the first cell, is now the end of the chain, and is

coupled to the rest of the cells from one side only. Again

using the electrical analogy, the overall impedance Z(2) of the

first two cells can be calculated as a function of the piezoelec-

tric diaphragm bulk modulus Bp(2) and the required density ra-

tio for this cell qeff(2). Following the same logic, the gains G(i)

for the different cells that would yield the prescribed set of

effective density values can be obtained. The impedance of

the ith cell and consequently the effective density and the

bulk modulus of the piezoelectric diaphragm for the ith cell

are hence given by

ZðiÞ¼

Zði�1Þ

�ZCDCðiÞþZ0pðiÞ

�Zði�1ÞþZCDCðiÞþZ0pðiÞ

0@

1AþZLDðiÞ

0@

1AZCFðiÞ

Zði�1Þ

�ZCDCðiÞþZ0pðiÞ

�Zði�1ÞþZCDCðiÞþZ0pðiÞ

0@

1AþZLDðiÞþZCFðiÞ

þZLFðiÞ;

(29)

qeffðiÞ ¼ ZðiÞHc

s lnr3ðiÞrlðiÞ

; (30)

BpðiÞ ¼Bf ln

r3ðiÞr2ðiÞ

Hl2dsZ0pðiÞcþ ðBp þ qdl2ds2Þ ln

r3ðiÞr2ðiÞ

� �þ s2qf ln

r2ðiÞr1ðiÞ

Hl2dl2

f sZ0pðiÞcþ ðBf l2d þ Bdl2

f þ qdl2dl2f s2Þ lnr3ðiÞr2ðiÞ

� �

s2 Hl2dl2f sZ0pðiÞcþ ðBf l2d þ Bdl2f þ qdl2

dl2f s2Þ ln

r3ðiÞr2ðiÞ

� �ln

r3ðiÞr1ðiÞ

:

(31)

The developed model for a system of four cascading cells

has been numerically verified with targeted relative density

values qr¼ 20, 0.05, 12, and 0.15. The frequency-dependent

bulk modulus and corresponding control voltages are as plot-

ted in Fig. 8.

IV. SENSITIVITY ANALYSIS AND PIEZOELECTRICSTABILITY

As mentioned earlier a shunted inductance in-parallelLp and capacitance in-series Cs are added to the piezoelectric

sector element to eliminate the instability that might occur

due to the applied control voltage to achieve the targeted rel-

ative density values. This instability might show up in the

form of negative value of the piezoelectric bulk modulus

indicating a state of buckling. This approach has proven to

be efficient as reported by Akl and Baz.27 However, adding

these two elements introduces an additional pole in the

frequency spectrum of the piezoelectric sector bulk modulus.

It is therefore the objective of this section to study the effect

of the added components on the operating bandwidth of the

newly developed composite acoustic metamaterial. The

expression for the bulk modulus Bp presented in Eq. (27) can

hence be rewritten as

Bp ¼ �qdl2ds2 � ðCp þ CsÞHl2ds2k/2

ðCp þ Cs þ CpCsLps2Þ ln r3

r2

þBf l

2ds2 qf ln

r2

r1

� qef f lnr3

r1

� �

l2f s2 qf ln

r2

r1

� qef f lnr3

r1

� �þ Bf ln

r3

r2

; (32)

which indicates the existence of two singularities; the first is

dependent on the cavity dimensions and properties and the

other is totally dependent on the electric components of the

TABLE 1. Parameters of acoustic cavity/piezodiaphragm system.

Parameter Value

H 0.05 m

c p/36 rad

R1 0.1 m

lf 0.01 m

ld 0.002 m

D �170� 10�12 m/V

qf 1000 kg/m3

qd 7800 kg/m3

Bf 2.25� 109 Pa

Bd 48.31� 109 Pa

er 1750



piezoelectric sector elements combined with the added in-

ductance and capacitance. The singularities (resonant fre-

quencies f1 and f2) occur at

f1 ¼1

2plf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBf ln

r3

r2

qf lnr2

r1

� qeff lnr3

r1

vuuuut ; (33)

f2 ¼1

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCp þ Cs

CpCsLp

s: (34)

Figure 9 illustrates a contour plot for the resonance fre-

quency of the electrical components of the acoustic metama-

terial as a function of Cs and Lp. In the design process, this

frequency has to be taken into consideration to select the val-

ues of Cs and Lp connected to Cp to resonate at a frequency

away from the targeted operating bandwidth. However, one

should be cautious reducing the values of Lp and Cs as this

significantly affects the control voltage needed, especially at

lower frequencies. Figure 10 shows the effect of Lp on VP.

Cs has shown no significant effect on the control voltage

needed to achieve a targeted relative density.

V. CONCLUSIONS

A new class of one-dimensional acoustic metamaterial

with controllable density, developed in cylindrical coordi-

nate system is presented. The proposed metamaterial is

designed to take a circular sector shape to allow for coupling

and/or integrating with irregular shaped objects. The effec-

tive density of the composite metamaterial has been theoreti-

cally proven to be controllable and thus any required value

of the effective density can be achieved along a very wide

FIG. 8. (Color online) (a) Piezodiaphragm bulk modulus and (b) control

voltage for four cascading and coupled cells targeting relative density values

qr¼ 20, 0.05, 12, and 0.15.

FIG. 9. (Color online) Electrical components resonance frequency as a func-

tion of Cs and Lp.

FIG. 7. (a) Schematic and (b) analogous circuit for a set of N cascading and coupled cells.



frequency spectrum. The theoretical analysis of this class of

active acoustic metamaterials is presented for an array of

water cavities separated by piezoelectric boundaries using a

lumped-parameter modeling approach. The same model is

also applied for a set of cascading coupled cells. A sensitiv-

ity and stability analysis is conducted to optimize the piezo-

electric electric parameters to prevent the instability that

might occur due to the applied control voltage. A natural

extension of this work is to include active control capabil-

ities to tailor the bulk modulus distribution of the proposed

metamaterial configuration.

ACKNOWLEDGMENTS

This work has been funded by a grant from the Office of

Naval Research (N000140910038). Special thanks are due to

Dr. Kam Ng and Dr. Scott Hassan, the technical monitors,

for their invaluable inputs and comments.

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FIG. 10. (Color online) Effect of LP on the control voltage Vp.



acoustic metamaterials with circular sector cavities and...

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